this stochastic
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R1.
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T
HE
A
STROPHYSICAL
J
OURNAL
, 536:331
»
334, 2000 June 10
2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.
(
INSTABILITY OF THE STOCHASTIC GALACTIC MAGNETIC FIELD
E. N. P
ARKER
Enrico Fermi Institute, University of Chicago, Chicago, IL 60637
AND
J. R. J
OKIPII
Department of Planetary Sciences, University of Arizona, 1629 E. University Boulevard, Tucson, AZ 85721-0092
Received 1999 November 3; accepted 2000 January 26
ABSTRACT
We examine the eÜects of the stochastic galactic magnetic –eld on the dynamical instability of the
interstellar gas and magnetic –eld. The large-scale random walk, or meandering, of the magnetic –eld
exerts stresses across the average magnetic –eld direction, which suppress the growth of perturbations
having a small wavelength normal to the –eld. The result is that, compared with a nonstochastic initial
magnetic –eld, those perturbations, which grow, are signi–cantly broadened in the direction normal to
the –eld. Hence, the instability in a stochastic magnetic –eld, such as that observed in our Galaxy,
should evolve into clouds that are more similar to those that are observed than are those found in the
absence of the stochastic –eld.
Subject headings:
ISM: magnetic –elds » instabilities » turbulence
1
.
INTRODUCTION
In a recent paper Kim et al. (1998) simulate the onset of a
dynamical instability in three dimensions under conditions
broadly similar to the gaseous disk of the Galaxy (Parker
1966). Their simulations treat an isothermal gas. Their
results show how the interstellar gas and magnetic –eld
evolve into a –nal nonlinear state consisting of closely
spaced thin vertical sheets of gas supported along the mean
magnetic –eld (as expected from linear analysis; Parker
1968a, 1968b), with rapid ambipolar diÜusion of the gas and
rapid reconnection between sheets. The gas eÜectively slips
downward relative to the escaping magnetic –eld (Parker
1968a, 1968b). They –nd that the gas accumulating in the
valleys along the –eld reaches a density of no more than
about twice the mean. They note that this relatively chaotic
state is diÜerent from the observed state of the interstellar
gas and magnetic –eld, where broad, massive gas clouds
accumulate with densities 10
2
or more times the ambient
value. They further comment that the sheetlike structure is
not consistent with observations of interstellar clouds.
We suggest in this paper that the next steps in their
numerical exploration of interstellar dynamics should be
1. to include a turbulent viscosity appropriate for the
interstellar motion of the order of 5 km s
~1
on scales of 10
2
pc, which will suppress the development of large wavenum-
bers perpendicular to the –eld,
2. to recognize the tendency of the interstellar gas tem-
perature in thermal equilibrium to decline with increasing
density, so that
d
p
/
p
B cdo
/
o
, where
c\
1, introducing the
thermal instability (Parker 1953a, 1953b) responsible for the
enormous gas densities in the interstellar molecular clouds,
and
3. to take account of the expected small-scale stochastic
topology of the galactic magnetic –eld (Jokipii & Parker
1968, 1969), which, as we show here, will also suppress large
wavenumbers perpendicular to the –eld.
Suggestions 1 and 2 are straightforward extensions of the
gasdynamics and will not be discussed further here.
2
.
INSTABILITY IN THE STOCHASTIC GALACTIC
MAGNETIC FIELD
We go on to consider in more detail suggestion 3, which
has not been discussed before in this context. The stochastic
wandering of the galactic magnetic –eld lines with corre-
lation lengths of the order of 10
»
100 pc (e.g., Jokipii &
Lerche 1969; Minter & Spangler 1996) eÜectively interlaces
all neighboring volumes of gas normal to the average mag-
netic –eld. Adjacent vertical and horizontal layers of gas
extending along the average –eld direction are, in fact, inter-
laced by the randomly wandering –eld lines, thereby inhib-
iting the up-and-down motion of adjacent slabs of gas and
–eld arising in the dynamical instability in the ideal, non-
stochastic –eld. That is to say, earlier speculations on rapid
diÜusion between these vertical slabs (Parker 1967; Lerche
& Parker 1968) seem to be ruled out by the stochastic
nature of the magnetic –eld. Furthermore, we expect that
one eÜect of the stochastic –eld will be to produce more
rounded clouds rather than the sheetlike structures pre-
viously envisioned. The idea is illustrated schematically in
Figure 1.
We consider the linear dynamical instability of a cold
plasma of density
o
(
z
) supported by magnetic pressure
against a uniform gravitational acceleration of the negative
z
-direction. The mean magnetic –eld
B
(
z
) extends in the
y
-direction, and barometric equilibrium requires that
d
(
B
2
/8
n
)/
dz
\[
o
(
z
)
g
. With the usual assumption that
B
2
P
o
, it follows that
B
(
z
)
\
B
(0) exp (
[
z
/
“
) and
o
(
z
)
\
o
(0) exp (
[
2
z
/
“
), where
“
is the characteristic scale
height
C
2
/
g
, where
C
is the characteristic speed
B
(0)/
Alfve
ç
n
[4
no
(0)]
1@2
.
We construct a crude model of a stochastic magnetic –eld
by introducing the horizontal transverse magnetic –eld
component
v
(
z
)
B
(
z
) in the
x
-direction, where
v
(
z
) is a small-
scale random function of
z
with zero mean and with
uniform statistical values. De–ne
a
2\
Sv
(
z
)
2
T
,
b
2\
S
(
d
v
(
z
)/
dz
)
2
T
. (1)
331
332 PARKER & JOKIPII Vol. 536
F
IG
. 1.»Schematic illustration of the eÜect of the meandering, stochas-
tic magnetic –eld lines on the development of the instability.
Top
: The
broadened peaks and valleys caused by the transverse stresses resulting
from the meandering, stochastic magnetic –eld.
Bottom
: The instability in
the absence of the stochastic magnetic –eld. Short wavelengths in the
direction normal to the magnetic –eld dominate the instability.
We take the correlation length of the —uctuating –eld to be
small compared with
“
and take
a
2
>
1 so that the zero-
order static magnetic pressure deviates but little from
B
(
z
)
2
/
8
n
. These assumptions are not strictly justi–ed by
observations of the present interstellar magnetic –eld
1
but
are used to simplify the analysis and do not detract signi–-
cantly from our main point concerning the eÜect of the
stochastic magnetic –eld on the instability. More realistic
systems can be considered numerically.
Perturb the system so that the
x
,
y
, and
z
components are
The
v
B
(
z
)
]
b
x
(
x
,
y
,
z
,
t
),
B
(
z
)
]
b
y
(
x
,
y
,
z
,
t
),
b
z
(
x
,
y
,
z
,
t
).
—uid velocity associated with the perturbation is denoted by
The linearized induction equations are
v
i
(
x
,
y
,
z
,
t
).
L
b
x
L
t
\
B
(
z
)
C
L
v
x
L
y
[
v
L
v
y
L
y
[
v
L
v
z
L
z
]
v
z
A
v
“
[
v@
BD
, (2)
L
b
y
L
t
\[
B
(
z
)
A
L
v
x
L
x
]
L
v
z
L
z
[
v
z
“
[
v
L
v
y
L
x
B
, (3)
L
b
z
L
t
\]
B
(
z
)
A
v
L
v
z
L
x
]
L
v
z
L
y
B
. (4)
1
We note that the most recent and detailed analysis of the observed
galactic magnetic –eld suggests that the random component is indeed
somewhat less (about .4) than the mean (Minter & Spangler 1996, 1997),
although earlier work suggested a larger random component (Heiles 1995;
Ruzmaiken, Shukurov, & SololoÜ 1988). Likewise, we note that our neglect
of the component of the initial random –eld in the
z
-direction is perhaps
more serious, since it aÜects the equilibrium»the gas can slide down the
randomly sloping magnetic –elds.
The linearized momentum equations are
o
(
z
)
L
v
x
L
t
\
B
(
z
)
4
n
C
L
b
x
L
y
[
L
b
y
L
x
]
A
v@
[
v
“
B
b
z
D
, (5)
o
(
z
)
L
v
y
L
t
\
B
(
z
)
4
n
C
v
A
L
b
y
L
x
[
L
b
x
L
y
B
[
b
z
“
D
, (6)
o
(
z
)
L
v
z
L
t
\
B
(
z
)
4
n
C
L
b
z
L
y
[
L
b
y
L
z
]
b
y
“
]
v
A
L
b
z
L
x
[
L
b
x
L
z
B
]
A
v
“
[
v@
B
b
x
]
B
(
z
)
“
]
v
A
v
“
[
v@
B
B
(
z
)
D
[
g
o
[
g
do
, (7)
where
do
is the density perturbation described by
Ldo
L
t
]
o
(
z
)
A
L
v
x
L
x
]
L
v
y
L
y
]
L
v
z
L
z
[
2
v
z
“
B
\
0 . (8)
The time-independent terms describe the zero-order static
equilibrium.
DiÜerentiate the momentum equations with respect to
time and use the induction equations and the continuity
equation to eliminate and
do
. The result is
b
i
L
2
v
x
L
t
2
\
C
2
G
L
L
y
C
L
v
x
L
y
[
v
L
v
y
L
y
[
v
L
v
z
L
z
]
A
v
“
[
v@
B
v
z
D
]
L
L
x
A
L
v
z
L
z
[
v
z
“
]
L
v
x
L
x
[
v
L
v
y
L
x
B
]
A
v@
[
v
“
BA
v
dv
z
L
x
]
L
v
z
L
y
BH
, (9)
L
2
v
y
L
t
2
\
C
2
G
[
v
L
L
x
A
L
v
z
L
z
[
v
z
“
]
L
v
x
L
x
[
v
L
v
y
L
x
B
]
v
L
L
y
C
[
L
v
x
L
y
]
v
L
v
y
L
y
]
v
A
L
v
z
L
z
[
v
z
“
B
]
v@
v
z
D
[
1
“
A
v
L
v
z
L
x
]
L
v
z
L
y
BH
, (10)
L
2
v
z
L
t
2
\
C
2
G
v
L
2
v
z
L
x
L
y
]
L
2
v
z
L
y
2
]
v
A
L
L
z
[
2
“
B
]
CA
L
L
z
[
1
“
B
v
z
]
L
v
x
L
x
[
v
L
v
y
L
x
D
[
v@
L
v
y
L
x
[
A
L
L
z
[
2
“
]
v@
B
]
C
L
v
x
L
y
[
v
L
v
y
L
y
[
v
A
L
v
z
L
z
[
v
z
“
B
[
v@
v
z
D
]
v
C
v@
L
v
y
L
y
]
v@
A
L
v
z
L
z
[
v
z
“
B
]
v@@
v
z
D
]
v
A
v
L
2
v
z
L
x
2
]
L
2
v
y
L
x
L
y
B
]
1
“
A
L
v
x
L
x
]
L
v
y
L
y
]
L
v
z
L
z
[
2
v
z
“
BH
. (11)
Then compute the local average over
z
by integrating
over
z
for a distance greater than the small correlation
length. The scale of variation of the perturbation is of the
1
随机过程
Written report of the attached
paper in
1300
word, Without the
equations
•
About
paper contents:
1
. The problem solved;
2
. Ideas and methods employed to solve the problem;
3
. Main conclusions.
•
Brainstorming (answer two questions at least):
1
. Is the problem proposed reasonable? Is it meaningful?
2
. Do the methods applied have any limitations? How to improve?
Are there any new solutions?
3
. What are the conditions for drawing your conclusion? How the
situation will be if the conditions are false? (make a bold guess
and give your reasons)
1
Stochastic Process
